Proportional Relationships
By the end of this lesson, you’ll be able to:
- Identify direct proportional relationships of the form \(y = kx\).
- Identify inverse proportional relationships of the form \(y = \frac{k}{x}\).
- Distinguish proportional relationships from non-proportional linear relationships.
- Interpret proportionality from equations, tables, and graphs.
- Find and use the constant of proportionality \(k\).
Key Ideas
A proportional relationship describes two quantities that change together in a predictable way.
There are two types:
- Direct proportionality: \(y = kx\)
- Inverse proportionality: \(y = \frac{k}{x}\)
These differ in how the variables change and what their graphs look like.
1. Direct Proportional Relationships
In a direct proportional relationship, one quantity is always a constant multiple of the other:
\[ y = kx \]
where:
- \(k\) is the constant of proportionality
- The ratio \(\frac{y}{x}\) is always the same
- The graph is a straight line through the origin (0,0)
Examples of Direct Proportions
- Cost \(=\) (price per unit) × (quantity)
- Distance \(=\) (speed) × (time)
- Recipe scaling
- Conversions (e.g., centimeters to inches)
Example:
If something costs $2.50 per pound:
\[ \text{Cost} = 2.5x \]
Here \(k = 2.5\).
Identifying Direct Proportionality
A relationship is directly proportional if:
- Equation is in the form \(y = kx\)
- Graph is a straight line and passes through \((0,0)\)
- Table has a constant ratio \(\frac{y}{x}\)
A relationship is not proportional if it has:
- a nonzero y-intercept (like \(y = mx + b\), \(b \neq 0\))
- a ratio \(\frac{y}{x}\) that changes
Example of not proportional:
\[ y = 2x + 3 \]
- Does not pass through the origin
- Ratio \(\frac{y}{x}\) is not constant
- Therefore not proportional, even though it’s linear
2. Inverse Proportional Relationships
In an inverse proportional relationship, one quantity increases while the other decreases:
\[ y = \frac{k}{x} \]
Characteristics:
- As \(x\) increases, \(y\) decreases
- The product \(xy\) is constant
- The graph is a hyperbola, not a line
- The graph never crosses the axes
Examples of Inverse Proportions
- Pressure and volume in physics
- Work-rate problems (more workers → fewer hours)
- Concentration/dilution
- Intensity vs distance (inverse-square variants)
Example:
If \(xy = 12\):
- If \(x = 3\), then \(y = 4\)
- If \(x = 6\), then \(y = 2\)
Both give the same product: \(12\).
Visual Comparison
(Insert the two visuals you generated: proportional vs non-proportional, and inverse proportionality)
- Direct proportional: line through origin
- Non-proportional linear: line with intercept
- Inverse proportional: hyperbola curve
Common Problem Types
1. Identifying Proportionality From a Table
Example:
| x | y |
|---|---|
| 2 | 10 |
| 4 | 20 |
| 6 | 30 |
Compute \(\frac{y}{x}\):
- \(10/2 = 5\)
- \(20/4 = 5\)
- \(30/6 = 5\)
Constant → directly proportional, \(k = 5\).
2. Identifying From a Graph
- Line through origin → direct proportion
- Line not through origin → linear but not proportional
- Curved hyperbola → inverse proportion
3. Writing an Equation (Direct)
Example:
A car travels at 60 miles per hour:
\[ d = 60t \]
\(k = 60\)
4. Writing an Equation (Inverse)
Example:
A job takes 24 worker-hours:
\[ xy = 24 \]
or
\[ y = \frac{24}{x} \]
5. Finding the Constant of Proportionality
- Direct: \(k = \frac{y}{x}\)
- Inverse: \(k = xy\)
Strategies
- Look for keywords:
- “per,” “each,” “every” → often direct proportional
- “inversely proportional,” “product is constant” → inverse
- “per,” “each,” “every” → often direct proportional
- Check table patterns:
- constant ratio → direct
- constant product → inverse
- constant ratio → direct
- Graph clues:
- line through origin → direct
- line not through origin → linear but non-proportional
- curve (like \(y = k/x\)) → inverse
- line through origin → direct
Worked Examples
Example 1 — Direct Proportion
Is \(y = 7x\) proportional?
Yes. It is of the form \(y = kx\) with \(k = 7\).
Example 2 — Not Proportional
Is the table proportional?
| x | y |
|---|---|
| 5 | 15 |
| 10 | 40 |
- \(15/5 = 3\)
- \(40/10 = 4\)
Not constant → not proportional.
Example 3 — Inverse Proportion
A quantity varies inversely with \(x\). When \(x = 4\), \(y = 8\).
Find the equation.
Product:
\[ k = xy = 4 \cdot 8 = 32 \]
Equation:
\[ y = \frac{32}{x} \]
Practice Problems
- Is \(y = 12x\) directly proportional?
- A relationship has \(xy = 18\). When \(x=3\), what is \(y\)?
- A table has points \((2,6), (4,13), (6,18)\). Proportional or not?
- Write an equation: “A plant grows 2.3 cm per day.”
- A graph passes through \((0,0)\) and \((3,12)\). Write the proportional equation.
- A quantity varies inversely with \(x\). If \(x=5\) and \(y=9\), write the equation.
1. Yes, directly proportional. \(k = 12\).
2. \(y = 18/3 = 6\).
3. Ratios: \(6/2=3\), \(13/4\neq3\) → not proportional.
4. \(h = 2.3d\).
5. \(k = 12/3 = 4\) → \(y = 4x\).
6. \(k = xy = 45\) → \(y = \frac{45}{x}\).
Summary
- Direct proportion: \(y = kx\), constant ratio, line through origin.
- Non-proportional linear: \(y = mx + b\) with \(b \neq 0\).
- Inverse proportion: \(y = \frac{k}{x}\), constant product, hyperbola.
- Direct and inverse proportionality appear in real-world rate, scaling, and work problems.
- Ratio constant → direct.
- Product constant → inverse.
- Origin matters for direct proportionality.